Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 7, Section 2 Completed
1
Volume of a Solid of Revolution: Using the Disk Method
Another application of the definite integral is the computation of the volume of a
particular type of three-dimensional soli
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 8, Section 8 Completed
1
b
1
Exercise 1: Use a TI-83 or TI-89 to evaluate 3 dx , where
1 x
(i.) b = 2
(ii.) b = 3
(iii.) b = 4
(iv.) b = an arbitrary positive integer
3
8
b
,
1
x
3
8
18
,
15
, .
32
dx =
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 9, Section 5 Completed
1
Alternating Series
Def.: Let ai > 0 for all i. An alternating series is a series that can be expressed in the form
or
1
i 0 3
Exercise 1: Show that
1
i 0 3
i
is an alternatin
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 9, Section 1 Completed
Sequences
Def.: A sequence is a
1
whose domain is the integers or a subset of the integers.
Note 1: For most of the examples that we will encounter, the domain of the sequence will
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 9, Section 4 Completed
1
In this section, we will continue to study convergence tests that apply to series with positive terms.
For the convergence tests developed so far in Chapter 9, the terms of the se
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 9, Section 8 Completed
1
In Section 9.7, we saw that Taylor polynomials can be used to approximate the values of a
function. In this section and in the next two sections, we will see that several importan
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 9, Section 3 Completed
1
In this section, we will study some convergence tests that apply to series with positive terms.
Integral Test
Theorem: If f is positive, continuous, and decreasing for x 1 , and i
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 8, Section 7 Completed
1
Revisiting Indeterminate Forms of Limits
f ( x)
0
by direct substitution, we get
. In
0 g ( x)
0
Calculus 1, we were occasionally able to genetically alter the molecular structure
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 9, Section 6 Completed
The Ratio Test
The Ratio Test is a test for absolute convergence.
Theorem: Let ai be a series with nonzero terms.
1.
a
i
converges absolutely if
2.
a
i
diverges if
3. The Ratio Test
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 9, Section 7 Completed
1
Linear Approximation (First Degree Polynomial Approximation)
Exercise 1: Determine the equation of the line tangent to the graph of f ( x) e x at x = 0,
and compare the linear app
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 7, Section 3 Completed
1
Volume of a Solid of Revolution Using the Shell Method
In this section, we will look at another method for determining the volume of a solid of
revolution, called the shell method
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 7, Section 1 Completed
1
Area of a Region Between Two Curves
Exercise 1a: Determine the area of the region between the two graphed lines, between
x = 4 and x = 12.
The region is trapezoidal, and the area
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 7, Section 5 Completed
1
Force
A force can be thought of as a push or a pull; a force changes the state of rest or the state
of motion of a body. For gravitational forces on the earth, for example, it is
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 7, Section 4 Completed
In this section, we will continue to find useful mathematical applications of the definite
integral. First, we will compute the length of a curve.
Length of a Curve
Exercise 1a: Sup
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 8, Section 4 Completed
1
Trigonometric Warm-Up Problem
Exercise 1: Determine an expression for each of the six trigonometric functions of .
sin( )
opp
hyp
3
cos( )
hyp
tan( )
2m
csc( )
sec( )
cot( )
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 8, Section 2 Completed
1
Integration by Parts
Integration by Parts is a technique of integration that is useful when the integrand
involves a product of an algebraic with a transcendental expression, such
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 8, Section 3 Completed
In this section, we will be looking at integrals which involve powers of trigonometric
expressions. Please try to contain your excitement.
Case 1: The Power of Sine is Odd and Posit
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 8, Section 5 Completed
1
1
dx using an old method. Strap on your helmet!
Exercise 1a: Integrate 2
x 2x 8
Now, x 2 2 x 8 = x 2 2 x
Note that
8 =
is of the form
( x 1) 2 3 2
In this case, u =
and a =
, so w
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 8, Section 1 Completed
1
Review of Basic Integration Rules
How many of the follow basic integration rules do you recall from Calculus 1?
1
du
u
n
du
=
u du
e
u
du
=
sin(u ) du =
u
du
=
cos(u ) du =
a
=
u
Bob Brown
CCBC Dundalk
Math 252 Calculus 2 Chapter 9, Section 2 Completed
1
Infinite Series
Def.: Let cfw_ai be a sequence. An infinite series (or, simply, a series) is the sum of the
terms of the sequence.
Note 1: Sometimes, it is convenient or necessary